2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse...

2x2 Matrices, Determinants and Inverses

1. Evaluating Determinants of 2x2 Matrices

2. Using Inverse Matrices to Solve Equations

1) Evaluating Determinants of 2x2 Matrices

When you multiply two matrices together, in the order AB or BA, and the result is the identity matrix, then matrices A and B are inverses.

10

01I

Identity matrix for multiplication


To show two matrices are inverses…

AB = I OR BA = I

AA-1 = I OR A-1A = I

Inverse of A Inverse of A

You only have to prove ONE of these.


Example 1:

Show that B is the multiplicative inverse of A.

17

13A

3.07.0

1.01.0B


Example 1:


17

13A

3.07.0

1.01.0B

3.07.0

1.01.0

17

13AB


Example 1:


17

13A

3.07.0

1.01.0B

3.07.0

1.01.0

17

13AB

10

01AB

AB = I. Therefore, B is the inverse of A and A is the inverse of B.


Example 1:


17

13A

3.07.0

1.01.0B

3.07.0

1.01.0

17

13AB

17

13

3.07.0

1.01.0BA

10

01AB

Check by multiplying BA…answer should be the same



Example 1:


17

13A

3.07.0

1.01.0B

3.07.0

1.01.0

17

13AB

17

13

3.07.0

1.01.0BA

10

01AB

10

01BA

Check by multiplying BA…answer should be the same



Example 2:

Show that the matrices are multiplicative inverses.

83

52A

23

58B


Example 2:

Show that the matrices are multiplicative inverses.

83

52A

23

58B

83

52

23

58BA

10

01BA

BA = I. Therefore, B is the inverse of A and A is the inverse of B.

The determinant is used to tell us if an inverse exists.

If det ≠ 0, an inverse exists.

If det = 0, no inverse exists. A Matrix with a determinant of zero is called a SINGULAR matrix



To calculate a determinant…

dc

baA dc

baA det



dc

baA dc

baA det

dc

ba Multiply along the diagonal



dc

baA dc

baA det

dc

ba

bcad

Take the product of the leading diagonal, and subtract the product of the non-leading diagonal

Equation to find the determinant


Example 1: Evaluate the determinant.

95

87det



95

87det

95

87det



95

87det

95

87

95

87det



95

87det

95

87

)5)(8()9)(7(

23

det = -23

Therefore, there is an inverse.

95

87det



24

24det



24

24det

)2)(4()2)(4( 0

24

24det



24

24det

)2)(4()2)(4( 0

24

24det

det = 0

Therefore, there is no inverse.


How do you know if a matrix has an inverse AND what that inverse is?

Given , the inverse of A is given by:

ac

bd

AA

det

11

Equation to find an inverse matrix

This is called the adjoint matrix. It is formed by interchanging elements in the leading diagonal and negating elements in the non-leading diagonal

dc

baA


Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M


Example 1:


45

22M

Step 1: Find det M


Example 1:


45

22M

Step 1: Find det M

)5)(2()4)(2( bcad

2

det M = -2, the inverse of M exists.


Example 1:


45

22M

Step 2: Find the adjoint matrix. i.e

ac

bd


Example 1:


45

22M

Change signs


ac

bd


Example 1:


45

22M

Change signs

?5

2?


ac

bd

Adjoint of M


Example 1:


45

22M

Change positions

?5

2?


ac

bd

Adjoint of M


Example 1:


45

22M


ac

bd

25

24

Change positions

Adjoint of M


Example 1:


45

22M

Step 3: Use the equation to find the inverse.

25

24

2

11M

ofMAdjoM

M intdet

11


Example 1:


45

22M

Step 3: Use the equation to find the inverse.

25

24

2

11M

15.2

121M


Example 2:


31

42


Example 2:


31

42

)1)(4()3)(2( bcad

2

31

42

31

42det


Example 2:


31

42

21

43

2

11A

15.0

25.11A

2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse...

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